13. Line Geometry in Terms of the Null Geometric Algebra over ℝ3,3, and Application to the Inverse Singularity Analysis of Generalized Stewart Platforms

Abstract

In this chapter, the classical line geometry is modeled in ℝ3,3, where lines are represented by null vectors, and points and planes by null 3-blades. The group of 3D special projective transformations SL(4) when acting upon points in space induces a Lie group isomorphism, with SO(3,3) acting upon lines.

As an application of the use of the ℝ3,3 model of line geometry, this chapter analyzes the inverse singularity analysis of generalized Stewart platforms, using vectors of ℝ3,3 to encode the force and torque wrenches to classify their singular configurations.